Bl .LETIN No. 10. »• !-• 

U. S. DEPARTMENT OF AGRICULTURE. 

DIVISION OF SOILS. 



S 



THE MECHANICS OF SOIL MOISIIE. 



BY 



LYMAN J. BRIGGS, 

PHYSICIST, DIVISION OF SOILS, 




WASHINGTON: 

GOVERNMENT PRINTING OFFICE, 
1897. 

d-iTfO-^'i 2.- 



3 



Bulletin No. 10. S. 12. 

U. S. DEPARTMENT OF AGRICULTURE. 
If 

DIVISION OF SOILS. 






THE MECHANICS (IE SIIIL MOISTERE. 



BY 



LYMAN J. BRIGGS, 

PHYSICIST, DIVISION OF SOILS 




WASHINGTON: 

GOVERNMENT PRINTING OFFICE. 
1897. 






LETTER OF TRANSMITTAL. 



IT. S. Department of Agriculture, 

Division of Soils, 
Washingtonj I>. C, Sejytemher 1, 1897. 
Sir : I have the honor to transmit herewith a paper upon theMechanics 
of Soil Moisture, prepared by Mr. Lyman J. Briggs, physicist of this 
Division. The subject is necessarily treated in a technical way in order 
that it may be clearly understood by the student of agricultural science. 
It exj)lains, however, more fully and clearl}' than ever before the actual 
cause of the capillary movement of water in soils and gives a much 
clearer knowledge of the laws and principles governing that movement 
than we have ever possessed. This is a subject of vast practical 
importance to the agriculturist, for the relation of his soils to water 
largely determines the class of crops which can be successfully grown 
upon them. 

This paper is a valuable contribution to science, and I recommend 
that it be published as Bulletin No. 10 of this Division. 
KesijectfuUy, 

Milton Whitney, 

Chief of Division. 
Hon. James Wilson, 

Secretary of Agrieulture. 



HOV 27 1906 



CONTENTS. 



Page. 

Introduction *> 

Properties of water atit'ectin!^' its rf't<'Uti«)ii and iiiovenient in the soil 6 

Gravitation of water 6 

Surface tension 7 

Viseo.sity 10 

HygToscopic state 11 

Properties of films 13 

Pressure of a film 14 

Surface of no pressure 15 

Form of water surface between two soil grains 18 

Establisbuient of equilibrium between two unequal masses of capillary water. 19 

Salts as affecting the movement of water in soils 20 

Temperature as aft'ecting the movement of water in soils 21 

Influence of texture and structure of soils on the acquirement and retention 

of soil moisture 21 

Disiilacoment of capillary water through gravitation 22 



ILLUSTRATIONS. 



Fio. 1. Condition of equilibrium among the surface tensions of three media in 

contact 9 

2. Pressure of a film 15 

3. Forms of the catenary 16 

4. Catenoidal film 17 

5. Cylindrical film 17 

6. Adjustment of water between two capillary spaces 19 

7. Displacement of capillary water through gravitation 23 

3 



THE MECHANICS OF SOIL MOISTURE. 



INTRODUCTION. 

It is intended in this bulletin to present the application of certain 
dynamical principles to the problems attending the movement and 
retention of soil moisture. Among these problems may be mentioned 
the capacity of a soil for water, the adjustment of water between a dry 
and a wet soil, the relation of texture, structure, and temperature to 
the water capacity, and the effect of fertilizers upon the water content 
of a soil. 

The extreme complexity of the texture and structure of soils renders 
very difficult any rigorous analysis of the phenomena connected with 
the movement of soil moisture. There are, however, two conceptions 
of the soil which have been jjroductive of good results. One is to con- 
sider the soil as made up of very fine particles without regard to the 
form of the particles or capillary spaces, the only condition being that 
the interstitial spaces are so small that the amount of interstitial space 
represented on any small section taken in any direction through the 
soil should be practically constant. Such a structure has been assumed 
by Dr. Katao^ as the basis of an extensive memoir on the movement 
of water in soils. In the other conception of the soil the particles are 
assumed to be of some simple geometrical form, such as the sphere. By 
means of different arrangements of these spherical particles, struct- 
ures having different amounts of interstitial space may be obtained. 
A calculation of the interstitial space for several arrangements has 
been given by Soyka,^ and a comparison with the structure of typical 
soils was made later by Professor Whitney.' 

While the principles that will be developed here are independent of 
the structure of the soil, the various arrangements of the soil grains 
which are set forth in the last form of structure referred to Avill be used 
to illustrate the principles and forces involved. 



' Ueber die Wasserbewegung in Boden. Bui. of College of Agriculture, Imperial 
University of Japan, vol. 3, No. 1, 1897. 
■^Forschungen auf dem Gebiete der Agrikultur-physik, B. 8, S. 1, 188."). 
■^Agricultural Science, vol. 3, p. 199, 1889. 



PROPERTIES OF WATER. AFFECTING ITS RETENTION AND M0VB3IENT 

IN THE SOIL. 

The water contained iu a soil may be considered to be of three kinds — 
gravitation water, capiUary water, and liygroscopic water. Gravitation 
water is that portion which is in excess of tlie amonnt wliicli the soil 
is able to retain nnder existing- conditions, and is consequently free to 
drain away. The capillary water is that part which would be retained 
in the capillary spaces under these conditions, and which is capable of 
movement through capillary action. The hygroscopic water is that 
found on the surface of the grains, which is not capable of movement 
through the action of gravity or capillary forces. 

The maximum amount of water which a given soil may contain 
depends upon the resultant effect of the two forces — gravitation and 
surface tension. The force due to gravity is proportional to the mass 
of the liquid considered, and is always directed vertically downward. 
In other words, it is the weight of the liquid. This nuiss of liquid would 
therefore leave the soil if not opi30sed by the action of some other force, 
the vertical component of which acting along the same line as the force 
of gravity must be equal to it and opposite in direction. The effective 
part of this force and the manner of its application, which is of the 
greatest importance in determining the movement of water in a soil, 
will be considered later. 

(iHA\ITATION OF WATER. 

When a column of soil is saturated with water, its lower end being 
left in such a manner that water can escape, a gradual draining of the 
soil takes place. The rate of flow of this water gradually becomes less 
and less until finally it ceases. The amount of water which thus leaves 
the soil under the ai^tion of the force of gravity would be gravitation 
water, while that remaining in the capillary spaces of the soil would be 
capillary water. 

There is no sharply-drawn line between these two quantities of water. 
The relative proportion depends, among other factors, upon the texture 
and structure of the soil, the surface tension of the soil water, the tem- 
l^erature, and the length of the column of soil considered. The impor- 
tance of this last factor can be shown from the following consider- 
ations: Suppose we have 100 cubic inches of soil packed into 100 cubical 
boxes without bottom or top, each containing 1 cubic inch. Suppose 
the soil in each box to be saturated with water. There will be a free 
water surface at the top and at the bottom of each box. By means of 
forces existing in these surfaces the water in each cube is enabled to 
overcome the attraction of gravity, so that each cube is able to retain 
an amount of Water equal to that necessary to produce saturation. In 
this case, therefore, there is no gravitation water. Suppose now that 
these cubical boxes are built up iu a vertical column 100 inches high. 
The water surfaces previously existing at the top and bottom, respec- 



tively, of two cubes now disappear when one cube is placed on top of the 
other. Instead of having- 200 surfaces as before, we now have only two 
surfaces, and they are called upou to support a column of water one 
huudred times as high as before. This they are unable to do, and 
water begins to drip from the lower surface. This water, which was 
previously what we have termed capillary, now becomes gravitational 
in its nature, due simply to a change in the length of the column. If 
the water in the soil was held in vertical capillary tubes running through- 
out the length of the column the water in each tube would simply fall 
until the two surfaces were able to support tlie weight of the liquid. 
In the soil, however, we have a different condition. As the water 
begins to leave the upper part of the column new surfaces are developed 
within the soil. As the water continues to drain away, these surfaces 
become more efficient in a way which will be explained later, and finally 
there comes a time when the opposing force exerted by these surfaces is 
sufficient to balance the weight of the liquid and the drainage ceases. 

As an example of the displacement of water toward the bottom of a 
vertical column of soil, it is of interest to consider some exj)eriments 
by Professor King,^ who showed that for coarse sands this displace- 
ment is very marked. A vertical tube, 42 inches in length, filled with 
sand, continued to discharge water at its lower end for forty days after 
saturation. At the end of that period the top G inches of soil contained 
but 2 per cent of water, while the bottom inches averaged 18 per 
cent. These results were in a general way corroborated by actual 
field observations on an area protected from ])recipitatiou and evapo- 
ration. No experiments were made on soils of a finer texture. It will 
be shown later that in such soils the movement would probably be 
much less marked. 

SURFACK TENSION. 

It has been pointed out that the force exerted on a liquid through 
gravity varies only with the mass of the liquid. This force can there- 
fore change in value only when the mass of the liquid varies. From 
this it follows that any movement of water which takes ])lace after 
equilibrium has been once established must be brought about through 
a change in the amonnt of water present in the soil or through a change 
in the force opposing gravitation. It is therefore of importance to con- 
sider the nature of this opposing force which we call surface tension. 

The phenomenon of surface tension is due to the existence of molecu- 
lar forces. In a suspended drop of Avater, for exami)le, the particles in 
the interior of the liquid are attracted equally in all directions by the 
other particles of the liquid. The resultant attraction on any particle 
in the interior is therefore zero, and it is free to move through the 
liquid. A particle on the surface of the drop, on the contrary, is not 
attracted equally on all sides, since the molecules of the gas surround- 



' Fluctnatioua in the Level and Rate of Movement ot Ground Water. F. H. Kiu< 
U. S. Dept. of Agriculture Bnl. No. 5, Weather Bureau, p. 25. 



8 

ing the drop exert less attraction upon tlie particle than is exerted by 
the particles of tlie liquid. The resultant attraction is therefore inward, 
along- a line i)eri)endicular to the surface of the liquid at that point. 
Now, the equations representing the behavior of the drop under the 
action of these forces are identical with those obtained if we imagine 
the drop inclosed in a water-tight membrane having a uniform tension. 
The action of the drop is therefore the same as if this imaginary mem- 
brane actually existed, and what we call surface tension is the tension 
that this ideal membrane would have to possess in order to produce the 
observed phenomena. This ideal membrane differs from all material 
membranes in that its tension does not change when the surface is 
increased. When the surface is extended, particles which were formerly 
in the interior are brought to the surface, so that the number of parti- 
cles per unit of area of the surface always remains the same. The sur- 
face tension is also practically independent of the form of the surface. 
The mathematical theory indicates a very slight increase where the 
mean curvature is concave, and a slight decrease where it is convex. 
This difl'erence is too small to be verified experimentally, but is of 
interest in connection with the fact that evaporation will take place 
from the convex surface at the same time that condensation is taking 
place on the concave surface. This in itself, then, must furnish a means 
of gradual adjustment of the water in the capillary spaces of a soil. 

It is of importance here to distinguish clearly between surface ten- 
sion and the effective force of a film. It has just been stated that the 
surface tension or the energy per unit area in the film is independent 
of tlie form and extent of the surface. The effective force or the pres- 
sure of the film, on the other hand, is dependent upon both the form 
and extent of the surface as well as upon the surface tension. This 
subject will be discussed more fully later. 

When a drop of a liquid is placed upon the horizontal surface of a 
solid, it either rapidly spreads out upon the surface in a thin film, as 
a drop of water on a clean glass plate, or else remains in the form of a 
drop, with as little surface as possible in contact with the solid, as in 
the case of mercury on glass. If we know the surface tensi(ms of the 
three surfaces which separate the solid and liquid, solid and gas, and 
liquid and gas, respectively, the action of the droj) can be anticipated. 
This arises from the following considerations: Let the solid, liijuid, and 
gaseous media in contact be represented by s, /, and //, respectively. 
The surfaces separating these three media will meet in a line. Through 
any point in this line pass a plane perpendicular to the line, and let 
this section be represented by the plane of the i)aper in fig. 1. 

At the point there exist three forces equal in magnitude to the 
tensions of the surfaces of separation and directed along lines tangent 
to the surfaces at that i)oint. In order that the system may be in 
equilibrium it is necessary that each of these forces shall exactly 
balance the resultant force of the other two. Let the vectors in the 



figure represent tlie direction and magnitude of these three forces. If 
these forces are in equilibrium, then lines drawn parallel to the vectors 
and equal to them in length will form the three sides of a triangle. 
The exterior angles of this triangle represent the angles between the 
surfaces of separation of the three substances. 

If a system is not in equilibrium, an adjustment must take place 
through change iu the direction of the vectors, since their magnitude 
remains constant. Consequently, the surfaces tend to change until 
the necessary angle is obtained. In general, if the tension of the solid- 
liquid surface is greater than the sum of the tensions of the other 
surfaces, the liquid will gather itself up in a drop, as in the case of 





Fio. 1. — Condition ef equilibrinni among the surface tensions of three media in contact. 



mercury. If, however, the solid-gas surface lias a tension greater than 
the resultant of the other two, the liquid tends to spread out over the 
surface of the solid. If the tension of the liquid-gas surface is greater 
than the difference between the tensions of the liquid-solid and the gas- 
solid, then the liquid finally reaches a condition of equilibrium. The 
angle between the liquid-solid and the liquid gas surfaces is known as 
the capillary angle. 

In the case of oil on water the tension of the water-air surface is 
greater than the sum of the other two tensions, so that no triangle of 
forces can be formed. The system is therefore unstable and the oil 
spreads out on the water indefinitely in a film until it ceases to have 
the same physical properties as the liquid. Lord Eayleigh has made 



10 

use of the minimum thickness of this film as the basis of au estimation 
of the maximum diameter of oil molecules. 

With the exception of mercury, water possesses a higher surface 
tension than any other substance which is liquid at ordinary tempera- 
tures. The surface tension of water • expressed in dynes per centi- 
meter is 75.0 at 0° 0. aud 72.1 at 25o. The temperature coefficieut is 
thus about — 0.14 dynes per degree Centigrade. The surface tension 
of most aqueous solutions of salt is higher than that of water, aud the 
surface tension increases with the concentration of the solution, as is 
shown in the following table: 

Surface tcusion of solutions of sails in ivaier. 



c ,. . ... T> -i Concen- Temper- Surface ten' 

Salt in solution. Density. ,.„t,i„„.„, at,„?«. sion. 



KCl... 
KCl ... 
KCl.... 
KaCl . . . 
NaCl... 
NaCl... 
K.,C0j . 

k;co3 

K.2CO, 
KNOi. 
KNO, 
MgSOj . 
MgSO, . 



170 
101 
046 
193 
107 
036 
357 
157 
040 
126 
047 
274 
068 



°C'. 

15-16 

15-16 

15-16 

20 

20 

20 

15-16 

15-16 

15-16 

14 

14 

15-16 

15-16 



Dynes per cm. 
82 8 
80.1 
78.2 
85.8 
80.5 
77.6 
90.9 
81.8 
77.5 
78.9 
77.6 
83.2 
77.8 



a Approximate weislit of the dissolved substance in 100 parts by weight of the solution. 

Most organic substances found in soils, especially those of an oily 
nature, being insoluble in water and hence most evident on the surface, 
lower the surface tension to a marked degree. The tension of soil 
extracts, therefore, is generally much lower than that of pure water, in 
spite of the presence of dissolved salts. 



VISCOSITY. 

It has been pointed out that the two great factors in determining the 
movement aud retention of soil moisture are gravitation and surface 
tension. We have now to consider a modifyiugintluehce which is exerted 
upon these factors through the viscosity, or internal friction, of the 
liquid upon which these forces are acting, the effect of which is to retard 
the establishment of equilibrium. The relative viscosity of Huids may 
be determined by their rate of flow through capillary tubes under uni- 
form conditions. Viscosity is generally expressed in terms of the coef- 
ticient of viscosity, which is numerically equal to the force necessary to 
maintain a flow of a layer of unit area past another layer of unit area 
with unit relative velocity. This coefficient is influenced by tempera- 
ture to a considerable extent. If we take tlie viscosity^ of water at 0° C. 



' Smithsonian Physical Tables, 1896, p. 128. These valnes are a mean of the results 
obtained Ijy Lord Rayleigh from the wave lenuth of ripples (Phil. Mag., 1890). aud by 
Hall from the direct measurement of the tension of a flat film (Phil. Mag., 1893). 

^Smithsonian Physical Tables, 1896, p. 136. 



11 

to be 100, the viscosity at 25° C. is 50, at 30° is 45, and at 50° about 31. 
This great variation iu viscosity with change of temperature is ilUis- 
trated iu the flow of water through soils, which Kiug ' found iu his leach- 
ing experiments but failed to explain. He observed the rate of flow at 
9° C to be 0.15 grams per minute, while the rate of flow at 32.5° C. was 
10.54 grams per minute. The ratio of the two rates of flow is 1.71. 
Now the viscosity of water at 0° C. as compared with water at 0° C. is 
75.G, and 32.5° is 42.5. The ratio of the two viscosities is 1.77, which 
iigrees very well with the ratio of the observed rates of flow. 

The viscosity of gases in opposition to that of fluids increases with 
increase of temperature. Air, which is largely used in making so-called 
'' permeability" determinations of soils, has a viscosity of 0.00017 
(14-.00273 t). An increase in temperature of 40° C. would therefore 
cause the coefficient of viscosity of air to increase one tenth of its 
amount. This evidently should always be taken into consideration in 
determining the physical character of a soil. G, Ammou,- in using air 
to determine the relative permeability of soils, and neglecting the 
change in viscosity with temperature, found that the permeability of a 
soil decreased with increase of temperature. The rate of flow of air 
at the higher temperatures as observed by him, when corrected for 
viscosity, agrees with the flow observed for the lowest temperatures 
within the errors of experiment. 

HYGROSCOPIC STATK. 

Most solid substances when exposed to ordinary atmospheric condi- 
tions condense upon their surfaces a slight amount of moisture. This 
moisture adheres with remarkable tenacity, and can be completely 
driven off only by prolonged heating at temperatures above the boil- 
ing point of water. In some soils the presence of hygroscopic moisture 
is very marked, on account of the large amount of surface presented by 
the soil grains. Air-dried samples, in which all visible evidences of 
moisture have disappeared, still contain under ordinary atmospheric 
conditions moisture in the hygroscopic form, amounting in some soils 
to 8 to 10 per cent of the dry weight. 

The table following, taken from a paper by Loughridge,-^ illustrates 
the variation of hygroscopic moisture in soils of different texture. 
These values were obtained by exposing the soil in a very thin layer 
to a saturated atmosphere, kept at a constant temperature, for a period 
of twenty-four hours. 

' U. S. Weather Bureau Bui. No. 5, 1892, p. 66. 

-Forschungeu auf clem Gebiete der Agriknitur-Physik, B. 3, S. 209. 
3 Investigations iu Soil Physics. — R. H. Loughridge. Report of the California 
Experiment Station, 1892-93, p. 70. 



12 



Hyfiroscopic moisture of noil 8. 



JS'ame and obaracter. 


HyKi-oscopic 
moisture. 

14. 5 
14.2 


Mechanical analysis. 


Chemical 


analysis. 


Clay. 

Per cent. 
32.6 
24.8 
52.2 
29.8 
31.5 
26.1 

12.1 
11.9 
10.5 

3.2 
2.6 
2.8 


Clay to 

25 mm. 

Per cent. 
74.0 
57.1 
67.9 
61.2 
76.8 
54.0 

55.6 
32.8 
34.9 

8.7 
8.9 
5.2 


Soluble 
silicates. 


Ferric 
hydrate. 

Per cent. 
7.7 
9.5 
29.7 
12.0 
9.1 


Clay: 

Black adobe 


Per cent. 
23.3 
54.0 
26.9 
34.3 
20.7 


lied nionutaiu soil 
Ked volt^aiiic soil . 


13 7 
11.1 
10.3 


Alkali soil 

Loams : 

Sediment soil 

Granitic soil 

Plains soil 

Sandy : 

(iila bottom soils . 


2.6 

9.2 
5.9 
4.9 

3.5 
1 2 


25.1 
21.0 
16.5 

8.9 


7.3 
6.2 
6.6 

7.4 


Do 


.8 











Under the conditions employed in tlio determinations given in the 
table it is not improbable that water was condensed in some of the 
more minute capillary spaces. Lord Kelvin ' has shown that such a 
minute capillary surface is capable of condensing moisture even when 
eva])oratioii is taking place from a neighboring plane surface of water. 
Some capillary spaces might therefore be able to hold minute quantities 
of water under conditions which would remove the water from larger 
spaces. In this way we might have some water held in a soil by capil- 
lary action, under conditions which would seem to indicate that the 
water (;ontent must be purely hygroscopic in its nature. 

The nature of this thin fllin which constitutes the hygroscopic mois- 
ture is not definitely known. It may extend uniformly over the surface 
of the grains independently of their form or nature; or it may be dis- 
continuous, occurring only in spots on the surface and depending to 
some extent on the form and nature of the grain. It would seem justi- 
fiable to assume that the amount of uioistuie thus held is proi)ortional 
to the surface of the grains, but this conclusion is supported only in a 
very general way by the results given in the investigation quoted, from 
which the above table was compiled. Loughridge remarks, in explana- 
tion of this, that the clays may be considered as very comi)lev substances 
made \x\) not only of particles in a very fine state of division, but com- 
bined with ferric, aluminic, silicic, and humic hydrates existing in the 
soil in greatly varied amounts. Even if the presence of tliese hydrates 
did not directly infiuence the hygroscopic water of a soil their decom- 
])Osition at the high temperature which it is necessary to maintain in 
order to drive oft" the hygroscopic moisture would introduce a disturb- 
ing factor in the value of the hygroscopic water content. On the other 
hand, it must be remembered that a mechanical analysis of a soil gives 
only in a very general way an idea of the surface area of the grains in 
a soil, and that a soil exposed to a saturated atmosphere is apt to 
acquire considerable moisture which is not strictly hygroscopic. The 



'Maxwell, Tlu-uiy of Heat, p. 287. 



13 

relation of hygroscopic moisture to the relative surface area j)reseutecl 
by diflerent soils, rteterDiined by a uietbod wbicli depends directly upon 
the amount of surface, will be made the subject of a later investigation. 

PROPERTIES OF FILMS. 

Since the movement and adjustment of water among tlie soil grains 
depend upon forces whose action is the same as if a uniform tension 
existed in the water surl'aces, it is very important to study the proper- 
ties of these surfaces and their effectiveness under difterent conditions. 
Water is not a convenient substance to use in the experimental study 
of tilms, since its high surface tension and low superficial viscosity 
make the film unstable, except when very small surfaces are used. An 
ordinary soap solution, on the other hand — or better, a solution of oleate 
of soda to which a quantity of glycerin has been added — gives films of 
great stability and highly adapted to experiment.^ 

The most familiar form of a soap film is the S])herical surface assumed 
by a bubble, as rei>vesenting the least surface area under the given con- 
ditions. The fact that the bubble will contract and finally become a 
plane fllin across the end of the i)ipe shows that there is a tension in 
the film which produces a pressure upon the air inside the bubble. 
Suppose, now, that a large and a small bubble are connected by means 
of a short pipe of circular cross section. It might be expected that the 
large bubble would blow the smaller one out until the two were of 
equal size. On the contrary, the opposite takes place. The small bub- 
ble contracts until it becomes a film across the end of the pipe. This 
film, however, is not a plane as in the first case, since on one side there 
is the pressure due to the large bubble and on the other the atmos- 
pheric pressure. It will consequently assume a spherical surface iden- 
tical with a segment of the large bubble, of which it in reality forms a 
part, separated from it by means of the pipe. 

The action of the smaJler bubble in blowing out the larger, or the 
pressure in the smaller bubble being' greater than in the larger, might 
have been to some extent anticipated if we had considered more fully 
the plane film across the end of the open pipe. Sup])ose such a film 
stretched across a circular ring of wire. This film might be looked 
upon as part of the surface of a great bubble of infinite radius. 
Since the pressure on both sides of the film is the same, it follows thnt 
in such a bubble the pressure inside would equal the pressure outside. 
It is not necessary, therefore, for the film to exert any pressure in either 
direction in order to maintain the equilibrium of the system. For a 
plane surface or a sphere of infinite radius it is evident that the pres- 
sure of the film is zero. 

Let us now consider a ring of smaller radius than the first, contain- 



' For particulars in regard to the preparation of the soap solution, see Soap Bub- 
bles, C. V. Boys, published by Young & Co., New York. This boolc also contains 
an account of many interesting and instructive experiments with iilms. 



14 

ing" a .segment of a bubble of such radius that the surface area of the 
segment is just equal to the surface area of the i)lane tilm held in the 
first ring. The surface energy is the same in the two cases. In the first 
case, however, it is directed in the j^laneof the ring, while in the latter 
case it can be divided into two portions, one of which acts in the plane 
of the ring and the other in a direction at right angles to this plane. 
This film, therefore, exerts a pressure, while the other does not, although 
both have the same energy. This pressure, therefore, evidently depends 
upon the form of the surface. This is of the greatest importance in 
connection with the problems relative to the movement of soil mois- 
ture. It is the form of the capillary surface which determines whether 
or not it is iu equilibrium with other water surfaces in the soil — 
whether it shall expand or contract, advance or recede. 

PHESSURE OF A FILM. 

It has been sliown that the pressure wliich a film is able to exert is 
dependent on the form of the film. It is now of interest and impor- 
tance to determine in what way the pressure is related to the curvature 
of the film and to its surface tension, in order tha< the action of films 
under certain conditions in the soil may be determined. 

Suppose a cylindrical film of radius r and pass a plane through it at 
right angles to the axis.^ Suppose the section to be represented by the 
plane of the paper. (See fig. 2.) Let ah be a small portion of the sec- 
tion of the film thus formed. Let T represent the surface tension 
directed tangentially at a and h in the plane of the section, the film 
considered being taken of unit width perpendicular to this plane and 
of length i(h. 

The resultant of the two components of T is cd or P. 

Then P=2Tsin ^ 6 

2r r 

T 
If ah=l, we have the unit area of surface and P= ; or the pressure 

varies directly as the surface tension and the curvature, the latter 
being the reciprocal of the radius. In general the pressure on any film 
at any point can be expressed by 

1 . 1 



V r, Vo I 



in which Tx and r-i represent the radii of curvature of the sections formed 
by passing two normal planes at right angles through the point. 

For a sphere ri=.r^, and we have P=^ or the pressure of a spherical 

surface is twice the surface tension divided by the radius. It is now 
easy to understand why the smaller bubble blows the larger one out. 

1 In what follows, but one surface of the film is consiflered. Where two concentric 
surfaces exist, as in bubbles, the pressure would be doubled. 



15 

Since the surface tension is the same on botli bubbles, if the smaller 
bubble is only one-half the diameter of the larj:;er its internal pressure 
would be twice as much. 



SURFACE OF NO PRKSSURE. 

It is evident from tlie equation that there are only two surfaces of 
revolution in which the pressure can be zero. For if we put P equal 
to zero it is necessary either that both /■, and r^ be equal to infinity, or 
that ri = —r, in order that the equation may be satistied. The first case 
is satisfied by the plane, which is a sphere of infinite radius. A plane 
film therefore has no pressure. The other case is satisfied by the 
catenoid surface in which ri = —r2, that is, in which the radii are numer- 
ically equal and on opposite sides of the film. This figure is of much 
importance in the consideration of films. It is generated by the revo- 
lution of the caten- 
ary curve {catena, ^ 
chain) about its di- 
rectrix. This curve 
is that represented 
by a chain or flexi- 
ble cord of uniform 
weight hanging 
from two horizontal 
suj^ports at a dis- 
tance from each 
other less than the 
length of the cord. 
The equation is: 



ij = ^!,[e--e ■') 




Pig. 2. — Pressure of a film. 



in which a is the 
distance from the 

lowest point of the curve to the directrix, which is taken to coincide 
with the axis of .r, and e is the base of the natural system of logarithms. 
It is evident that the curve is symmetrical about the axis of y and that 
its intercept on the axis of y is equal to the value of a chosen. This 
curve is illustrated in fig. 3, in which different forms are obtained bj^ 
assigning different values to a. They represent the different ])ositions 
assumed by a chain under different tensions, or by a film having different 
surface tensions. 

One of the best examples of the catenoid is found in a soap film 
stretched between two equal circular rings of wire so placed that their 
planes are i)arallel and their edges bound a rectilinear cylinder. This 
can be easily prepared by wetting the rings in the soap solution and 
blowing a bubble upon them. The film inside each ring is then broken 
with a hot wire, after which the rings may be separated. Since the 



16 

pressure is now tbe sauie on both sides of the tiliii we have a surface 
of uo pressure. This surface is stable only wheu the tangents to the cate- 
nary at its extremities intersect before reaching the axis of the figure. 
The accompanying' illustration (flg. 4) is taken from a photograph of 
a film prepared in the manner described. 

A section of the catcnoid midway between tbe wire rings and parallel 
to them is a circle. A circle of this diameter if cut out from a piece of 
paper aud held with its edge against the catenary would exactly fit the 
surface for a short distance each side of the center, or the surface is one 
of no curvature. Since the potential energy of any system always tends 
to reach its minimum, the film tends to contract to its smallest possible 



8 












Y 












\ 




1 














1 




1 


7 

6 

5 

4 

3 

2 
1 




\ 


I 














/ 


1 






\ 
























A 


\ 










/ 


/ 






^ 


\\ 








\ 


u 


'/ 








\ 


^ 


►^ CLz 


5^^ 


j 


7 












^ 


'^«~...^; 


2 ^ 
















\ 


\sct= 


-J 




















\^ 











o 

Fig. 3, — Foriius of tlie caienary . 



area and the film between the two rings represents the surface of least 
area which can be formed between them. 

Let a small soap bubble be blown between two rings as in the prepa- 
ration of the catenoid soap film. Now separate the rings until the film 
between the rings forms a cylinder with straight walls, as in fig. 5. 
The pressure inside the cylinder is evidently greater than the atmos- 
pheric pressure, as is shown by the bulging outward of the films inside 
the rings. In -fact, from a measurement of the curvature of these 
spherical segments at the end of the cylinder, we find the pressure in 
the cylinder to be equal to that of a spherical bubble of twice the 
diameter of the cylinder. 



17 




Xow insert the i)ipe in the cylinder through one of the ends and allow 
the pressure inside and out to become equalized. The hlms inside the 
rings become plane, since the plane is a surface of no pressure. The 
cylinder degenerates into a catenoid, also a surface of no i)ressure. 
Suppose now the pressnre is reduced still more, so that the films inside 
the rings bulge inward, showing the pressure on the inside to be less 
than outside. This gives a surface having a more pronounced waist 
than the cate- 
noid. It repre 
seuts the form 
assumed by the 
free surface of a 
liquid between 
two solid sur- 
faces, which it 
wets and is 

.1 ,. n 1 Fin. 4.— Catenoidal film. 

thereiore called 

a capillary surface. This form of surface is therefore the one that is 
met with in the capillary si)ace formed by the contact of two soil 
grains. The generating curve of a capillary surface is known as the 
"elastic curve,'' from its identity with the curve formed by a straight 
spring of uniform tlexibility when its ends are acted upon by equal and 
opposite forces. Some of the forms assumed by this curve under dif- 
ferent conditions can be easily obtained with a steel spring.^ 

To summarize briefly, the 
pressure which a film may 
exert dej)ends upon its sur- 
face tension and its form: 
(1) The pressure of the film 
may be inward, as in the case 
of the cylinder, indicated by 
the films inside the rings 
being forced outward; (2) 
the pressure may be zero, as 
in tlie case of the catenoid, 
the films inside the rings 
being plane; (3) the pre-ssure may be outward, as in the case of surfaces 
generated by the elastic curve, the hlms within the ring being drawn 
inward. 

If the interior of two closed films of unequal size, both having a pres- 
sure inward and both free to move, are connected, the smaller film hav- 
ing the greater pressure will contract, forcing the larger film to expand. 
This is illustrated by two soap bubbles or by the coalescence of drops 
of mercury. If the two films have a pressure outward, the smaller film. 




-rvliiiilric;il film. 



' For traciugs ol)taiue(l in this way sei! Thomson & Tait, Natural Philosophy, 
Part I, p. 148. 

7293— Xo. 10 2 



18 

having the greater cnrvature, will expand until the two films become 
equal. This is what would take place in surfaces generated by the 
elastic curve — that is, capillary surfaces — if these surfaces were free to 
move, and is of the greatest importance in the adjustment of the water 
content of a soil among the capillary spaces. 

FORM OF AYATEll SURFACE BETWEEN TWO SOIL GRAINS. 

The manner in which water is held in a soil may now be considered. 
Suppose the soil grains to be momentarily separated so that no two 
grains are in contact. Each grain carries with it a small amount of 
moisture, which, through the agency of surface tension, spreads over 
the surface of the grain. Suppose now that two of these grains, which 
may be assumed spherical in form, are once more brought in contact. 
The water is drawn into the capillary space fornuMl between the spheres 
and forms between the grains a capillary water surface. 

The collection of a portion of the water previously distributed over 
the surface of the grains into this capillary space may be e\i)lained 
from a consideration of the curvature and pressure of the water sur- 
faces, which will hereafter be designated as films. When the two 
spheres are brought in contact, the films in meeting form a surface 
of very great curvature having a pressure outward. Since the two 
films on the surface of the spheres have a pressure inward, the water 
moves rapidly toward the capillary space. As the new surface increases, 
its pressure becomes less and the ijiovement of water becomes slower, 
finally ceasing when the pressure is not sufficient to overcome the 
resistance oiiered b^' the films. 

The arrangement of the water between the spheres may also be con- 
sidered from the standpoint of the potential energy of the system. It 
has been shown that a film between two rings assumes the form of the 
catenoid, as representing the least surface, and consequently the mini- 
mum potential energy. Therefore, any liquid surface held between two 
parallel planes will tend, so far as the conditions will admit, to ap))roach 
the catenoid in form, which is the limiting form of the capillary sur- 
face. This is well illustrated by the ink between the blades of a right- 
line ruling pen. These parallel planes may be supposed to separate 
gradually as the amount of water increases, and this gives the condition 
on the spherical surfaces. 

On the assumjition that no evaporation can take place and that the 
spheres are entirely covered with moisture before being brought in 
contact, it follows that the spheres will still be covered after contact, 
although the film will be much thinner than before. The amount of 
diminution in the surface, due to the contact of the S])heres, is evidently 
ecpial to the difference between the combined area of the original water 
surfaces and the combined area of the spherical segments outside the 
ca])illary surface, together with the area of (he capillary suiface. If 
tlie thickness of the original fihns on the spheres was small in compari- 



19 

sou with their radii, the diminntiou in surface is practically equal to the 
diHereuce betweeu the combined areas of the spherical surfaces inclosed 
in the capillary space and the area of the inclosing capillary surface. 
If the equatiou of the capillary surface, which is too complex to be 
considered profitably here, is known, the area of the surface cau be 
calculated aud the change in potential energy determined. 

ESTABLISH3IEjNT OF EQUILIBRIUM BETWEEN TWO UNEQUAL MASSES 

OF CAPILLARY WATER. 

Suppose the capillary spaces formed by several spheres in contact to 
contain different amounts of water. These spheres are sup[)osed origi- 
nally to have beeu covered with films aud then brought in contact, 
so that water films will exist ou all surfaces which are not submerged. 
Let oue of these films connect the water held in two adjacent capillary 
spaces containing difierent quantities of water. This is illustrated 




Fig. 6.— Atljustiucnt of water between two capillarj' spaces. 

diagrammatically in fig. C, which represents a section through the 
centers of three spheres in contact. 

Since the capillary s])aces are similar in form, it follows from previous 
considerations that the surface of the smaller mass of water will have 
the greater curvature, and consequently the greater pressure outward. 
The direction and relative magnitude of the pressures of the two films 
taken within the section are I'epresented by the length of arrowvS. 
Since the surface of the lesser mass of water exerts the greater pres- 
sure outward, water will move through the connecting film in the 
direction of the curved arrows from the greater to the lesser mass. 
This action will continue until the pressure becomes the same, which 
in this case takes place when the capillary spaces contain equal amounts 
of water, since the spaces are of the same form. In any case, equi- 
librium is reached when the two films attain the same curvature. 

The rate at which this adjustment of water between two capillary 



20 

spaces will take place depends upon tLe Aiscosity of the coiiiieating 
film, the surface teusion, and tbe difference in curvature of the films. 
The Aiscosity of the connecting film does not in any Avay interfere with 
the final adjustment of the water, but it retards to a greater or less 
degree the establishment of equilibrium. An in(;rease in either the 
enrvature or the surface tension causes an increase in pressure, as has 
been previously pointed out. 

It is evident that this movement can be extended to any number of 
capillary spaces through any number of films, so that adjustment takes 
place over a large mass of soil wheii disturbing influences are intro- 
duced. This change takes place more or less slowly, according to tbe 
amount of water present in the soil If the soil is nearly saturated, so 
that the films connecting the capillary spaces are short and thick and 
the capillary spaces themselves are not active, but little resistance is 
offered to the movement of water and the addition of water at the sur- 
face is quickly felt farther down. If, on the other hand, the soil con- 
tains hut little water, the same amount of water added to the surface, 
while producing marked changes in the upper layers, will not be felt 
so quickly at the lower depths on account of the activity of the upper 
capillary spaces and the length and small cross section of the con- 
necting films. But an adjustment of the water between the upper and 
lower capillary spaces takes x>bace in this case also until equilibrium 
is gradually reached. 

SALTS AS AFFECTUM G THE MOVEMEJN^T OF WATER IN SOILS. 

We have seen the importance of surface tension in opposing the, 
gravitation of capillary water of soils. Any change in the surface 
tension of the soil moisture tends to bring about an adjustment of the 
water throughout the whole mass of soil. If the surface tension of 
the water in the upper layers of a soil is increased, water is drawn 
toward that point. Since the surface tension of most salt solutions is 
higher than that of water, and the surface tension increases with the 
concentration of the solution, it might be expected that any salt used 
as fertilizer a solution of which has a high surface tension would 
increase the amount of water in the soil. 

It must be remembered, however, that the surface tension of solu- 
tions is very greatly decreased by the addition of very small quantities 
of certain organic substances produced through the decomposition of 
vegetable matter. This action is especially marked where there are 
present substances of an oily nature which do not go into solution, but 
spread out over the surface in an extremely thin film. Owing to such 
substances being continually produced by the decay of organic matter, 
the surface tension of the soil moisture is kept very low and could be 
only slightly influenced by the addition of salts. The application of 
substances to the soil for the purpose of changing its water content 
through a change in the surface tension, w(mld not therefore neces- 
sarily be productive of marked results. 



21 

TEMPERATURE AS AFFECTING THE MOVEMENT OF WATER IN SOILS. 

It was pointed out in a preceding section that the surface tension of 
water decreases witlj increase of temperature. Therefore if the bottom 
of a column of soil in which the water has attained a condition of 
equilibrium should be cooled, the surface tension of the lower strata 
would be raised and the water would be drawn toward the bottom, or 
if the lower strata should be heated the water would tend to move 
toward the top. The tirst method of i^rocedure should give the most 
marked results, since the movement in this case is assisted by gravita- 
tion. A movement should also be secured by raising or lowering the 
temperature of the whole mass of soil uniformly. In the first case the 
water content of the upper strata would be decreased and in the second 
case increased. These conclusions are indirectly verified by some inter- 
esting experiments of Professor King ' in experimenting with the fluc- 
tuations of ground water in a large, cylindrical, galvanized iron tank. 
He found that the water in a circular well in the middle of the cylin- 
der rose daily and. fell again during the night. The application of cold 
water to the outside of the cylinder by means of a hose also caused 
the water in the well to fall. These results are fully consistent with 
the phenomena of surface tension. When the temperature of the soil 
was raised the surface tension of the water was lowered and more 
water was drawn into the lower part of the cylinder, Avhich raised the 
level of the water iu the well. When cold water was applied to the 
outer surface of the cylinder, the water m tiie soil was drawn up again 
tlirough increased surface tension and the level of the water in the 
well was lowered. 

The influence of temperature on the rate of flow of water iu satu- 
rated soils is very great. This is due to a change m the viscosity of 
water with temperature, as has been pointed out in the section on vis- 
cosity. This property is not only of interest in considering saturated 
soils, but it is also an important factor in determining the rate of 
adjustment of water in soils in which saturation is not complete. 

INFLUENCE OF TEXTURE AND STRUCTURE OF SOILS ON THE 
ACQUIREMENT AND RETENTION OF SOIL MOISTURE. 

The limit of the capacity of any soil for water is reached when the 
surface tension holding the water in the capillary spaces is no longer 
able to overcome the force of gravity acting on the mass. Tlie relative 
water capacity of two soils, therefore, depends principally upon the 
number and size of the capillary spaces. By a capillary sjiace as used 
here is meant not any interstitial space in the soil structure, but only 
that portion of it which is near the point of contact of two soil grains. 
It is that portion in which the bounding walls are close together, sepa- 
rated only by distances of capillary magnitude and consequently most 
efficient iu retaining water. It is evident that in a soil of fine texture 



' U. S. Department of A.uiicT-ilt;!ie Wn]. No. 5. Weather Bureau, pji. 59-01. 



22 

the grains luiglit b'e so close together as to make all the interstitial 
«j)ace capillary in its nature. 

The one important factor which determines the acquirement and 
retention of soil moisture is the curvature of the capillary water sur- 
faces. If ecpial volumes of two soils are placed in contact, and the 
curvature of the surface is less iu the first than in the second, then 
water will move from the first to the second, increasing the curvature 
in one and decreasing it in the other until it becomes the sam^in both 
soils. If the second soil contains a greater number of capillary spaces 
than the first, it will contain more water when equilibrium is estab- 
lished. During the adjustment water will have actually moved from a 
soil containing a low i)ercentage of water to one having a higher i)er- 
centage. In no case, however, will water leave a capillary space having 
a water surface of large curvature to go to a space with a surface of 
less curvature. It is the form of the surface which determines the 
movement of the water. 

In a form of structure presented by Dr. Soyka, to which reference has 
jireviously been made, and iu which the spheres are arranged for the 
greatest amount of interstitial space, there are only about one half as 
many points of contact between the grains as in another form of struc- 
ture given, although the amount of interstitial space in the first case is 
twice as great. Consequently the second form — the couipact soil — 
would have twice the water capacity of the first, since the number of 
capillary spaces formed is twice as great. The difference in this case 
would be due entirely to the structure of the soil, since the texture 
remains uniform. 

In.the same manner a soil of fine texture contains many more capil- 
lary spaces than a soil which is coarse, and consequently has a much 
greater water holding power. In a coarse sandy soil the interstitial 
spaces are large, allowing percolation and drainage to take place rap- 
idly, but permitting the formation of comparatively few capillary spaces 
for the storing of water. As the texture becomes finer the interstitial 
sjjace becomes smaller and the capillary spaces increase in number, and 
embrace a large proportion of the whole interstitial space. The water 
capacity of the soil increases and percolation is greatly decreased. 
The limit is reached when the texture becomes so fine and the structure 
so close that all the interstitial space becomes capillary in its nature. 
The capacity of the soil for moisture iu this case is reached only when 
all the interstitial space becomes filled with water, a condition found in 
some clay soils. 

DISPLACEMENT OF CAPILLARY WATER THROUGH GRAVITATION. 

In considering the adjustment of water among the capillary spaces 
and the arrangement of the film so as to present as little free space as 
possible, it was assumed that the action of gravity could be neglected. 
This is undoubtedly the case in the smaller capillary spaces, but in 
those existing between the larger soil grains gravitation causes a dis- 
placement of the capillary water, so that it is no longer symmetrically 



23 






i < ' f 



arranged about tlie point of contact. The action of gra\ ity on the 
capilhiry water between two spheres is ilhistrated in lig. 7. 

The model consists of rnbber balls about 1 inch in diameter fas- 
tened together by small steel pins inserted normal to the surface at the 
points of contact. The whole is then immersed in cylinder oil and 
iillowed to drain while in a vertical ijosition. While the great size of 
the balls and the low surface tension of the liquid are conditions not 
found in soils, it nevertheless illustrates the principle involved. If the 
]i(pud had no weight it would be uniformly distributed about the line 
of centers of spheres. Gravitation, however, causes a distortion of the 
liquid from a position of symmetry about the point of contac-t. When 
the line of centers is horizontal, the liquid moves down until the pres- 
sure exerted by the upper x^ortion of the tilm is equal to that exerted 
by the lower portion plus the weight of the liquid. The curvature of 
the upijer part of the film must therefore be 
greater than that of the lower. The upi)er 
film is consequently drawn down into the 
capillary si^ace until it acquires sufficient 
curvature to support the weight of the liquid 
and the, tension of the opposing film. Ref- 
erence to the figure will show the upper part 
of the film far down in the capillary siiace, 
while the lower part is well down on the sur- 
face of the spheres. 

If the line of centers is vertical the curva- 
ture from the upper contact will gradually 
increase downward in order to compensate 
the effect of the weight of the liquid. This 
can be seen in the vertical films in thefigure, 
and is well illustrated by the position as- 
sumed by a drop of ink between the opened 
blades of a ruling pen, held so that the blades lie in h(nuzontal i)lanes 
one above the otlier. 

A consideration of the displacement of the water in the capillary 
spaces through the influence of gravity serves to give an idea of what 
takes place when the soil contains more water than the capillary spaces 
are able to retain. The upper part of the film is then unable to reach 
a position where it (!an balance the opposing forces, and water is moved 
through gravitation until the film is enabled to secure a position in 
which it can establish equilibrium. Such displacements of capillary 
water would occur only in the larger capillary spaces where a consider- 
able mass of w^ater would be subjected to the action of gravitation. 

In considering the gravitation of water in soils, mention was made 
(p. 7) of the marked difference in water content of a vertical column 
of soil at different points along the column. Having developed the 
principles which regulate the movement of water in soils, the exijlana- 
tion of this non-uniform distribution of water in a vertical column can 
now be given. 






Fig. 7.— Displacement of capillary- 
water through gravitation. 



^ 



24 



Consider a single vertical column of soil grains as arranged in fig-. 7, 
and suppose the capillary spaces to contain equal amounts of water. 
The amount of water in each capillary space is assumed to be less than 
the amount required to saturate it considered by itself, so that if the 
column were in a horizontal position the water would all be retained. 
Each capillary surface would in tins case be supporting the water in 
the capillary space, together with the weight of the water in the con- 
necting film on the surface of a soil grain. When the column is raised 
to a vertical position, the weight of the whole conducting film from top 
to bottom is thrown for an instant upon the capillary surface nearest 
the top. This capillary space immediately loses water until the pres- 
sure of the surface is equal to the pressure of the surface next below, 
plus a pressure sufticient to balance the weight of the water in the con- 
necting film between the two surfaces. Both spaces then lose water 
together, maintaining this difference in pressure until the pressure of 
the second surface is equal to that of the third plus the pressure neces- 
sary to balance the weight of tlie connecting film, as before. The j)res- 
sure of the first surface Avould now be equal to the pressure of the third, 
plus the pressure necessary to balance the weight of the two connect- 
ing films. The action would continue in this manner through each 
capillary surface until e(pulibriuui is established. 

The relative part taken by the different capillary surfaces in support- 
ing the weight of the connecting column can be illustrated by means of 
the following mechanical analogue. Suppose a series of very thin elas- 
tic membranes stretched over circular hoops and supi)orted horizon- 
tally one above the other, the distance between the membranes being 
very small. Now, let a heavy ball be placed upon the upper membrane. 
This membrane immediately stretches until it reaches the second, then 
the two stretch together until they touch the third, and so on, until the 
ball comes to rest. Suppose the membranes all had the same tension 
before the ball was piit in place. After equilibrium is established the 
upper membrane will have the greatest tension, the second one the next 
greatest, and so on, analogous to the pressure of the capillary surfaces. 

The pressure of the upper capillary surface will always exceed that 
of the lower surface by the pressure necessary to support the weight 
of the conducting column. This would necessitate the upper films 
having a much greater curvature, so that less water could be held in 
the capillary spaces. The water content should therefore increase 
uniformly from top to bottom. This has been shown to be the case 
with coarse sands. 

In a soil of fine texture the number of capillary spaces is greatly 
increased. The pressure exerted by the capillary surfaces would there- 
fore be mucli greater for the same water content. Consequently, the 
effect of the weight of the connecting films would be much lessened, and 
the water content of the soil would be much more uniformly distributed. 

O 



7h.A 



LBJe'07 



i 



i 



